Hostname: page-component-54dcc4c588-2bdfx Total loading time: 0 Render date: 2025-10-04T05:50:44.991Z Has data issue: false hasContentIssue false
  • English
  • Français

LYNDON INTERPOLATION PROPERTY FOR EXTENSIONS OF $\mathbf {S4}$ AND INTERMEDIATE PROPOSITIONAL LOGICS

Part of: Model theory General logic

Published online by Cambridge University Press:  12 August 2025

TAISHI KURAHASHI
TAISHI KURAHASHI*
Affiliation:
GRADUATE SCHOOL OF SYSTEM INFORMATICS https://ror.org/03tgsfw79 KOBE UNIVERSITY 1-1 ROKKODAI, NADA, KOBE 657-8501, JAPAN
*
E-mail: kurahashi@people.kobe-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the Lyndon interpolation property (LIP) and the uniform LIP (ULIP) for extensions of $\mathbf {S4}$ and intermediate propositional logics. We prove that among the 18 consistent normal modal logics of finite height extending $\mathbf {S4}$ known to have CIP, 11 logics have LIP and 7 logics do not. We also prove that for intermediate propositional logics, the Craig interpolation property, LIP, and ULIP are equivalent.

Keywords

interpolation propertiesmodal logicintermediate logics

MSC classification

Primary: 03B45: Modal logic (including the logic of norms) 03B55: Intermediate logics 03C40: Interpolation, preservation, definability

Information

Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 33
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Chagrov, A. and Zakharyaschev, M., Modal Logic , volume 35 of Oxford Logic Guides, The Clarendon Press; Oxford University Press, New York, 1997. Oxford Science Publications.Google Scholar
Craig, W., Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory . The Journal of Symbolic Logic , vol. 22 (1957), pp. 269–285.Google Scholar
Fitting, M., Proof Methods for Modal and Intuitionistic Logics , volume 169 of Synthese Library, D. Reidel Publishing Co., Boston, 1983.Google Scholar
Gabbay, D. M., Semantic proof of Craig’s interpolation theorem for intuitionistic logic and extensions. II , Logic Colloquium ’69 (Proceedings of the Summer School and Colloquium in Mathematical Logic, Manchester, 1969) (R. O. Gandy and C. M. E. Yates, editors), volume 61 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1971, pp. 403–410.Google Scholar
Gabbay, D. M. and Maksimova, L. L., Interpolation and definability, volume 46 of Oxford Logic Guides, The Clarendon Press; Oxford University Press, Oxford, 2005.Google Scholar
Kurahashi, T., Uniform Lyndon interpolation property in propositional modal logics . Archive for Mathematical Logic , vol. 59 (2020), nos. 5–6, pp. 659–678.Google Scholar
Kuznets, R., Proving Craig and Lyndon interpolation using labelled sequent calculi , Logics in Artificial Intelligence (L. Michael and A. Kakas, editors), volume 10021 of Lecture Notes in Computer Science, Springer, Cham, 2016, pp. 320–335.Google Scholar
Kuznets, R., Multicomponent proof-theoretic method for proving interpolation properties . Annals of Pure and Applied Logic , vol. 169 (2018), no. 12, pp. 1369–1418.Google Scholar
Kuznets, R. and Lellmann, B., Interpolation for intermediate logics via hyper- and linear nested sequents , Advances in Modal Logic (G. Bazhanishvili, G. D’Agostino, G. Metcalfe, and T. Studer, editors), 12, College Publications, London, 2018, pp. 473–492.Google Scholar
Kuznets, R. and Lellmann, B., Interpolation for intermediate logics via injective nested sequents . Journal of Logic and Computation , vol. 31 (2021), no. 3, pp. 797–831.Google Scholar
Lyndon, R. C., An interpolation theorem in the predicate calculus . Pacific Journal of Mathematics , vol. 9 (1959), pp. 129–142.Google Scholar
Maksimova, L. L., Craig’s theorem in superintuitionistic logics and amalgamable varieties of pseudo-Boolean algebras . Algebra i Logika , vol. 16 (1977), no. 6, pp. 643–681. 741. English translation in: Algebra and Logic, 16(6), 427–455, 1977.Google Scholar
Maksimova, L. L., Interpolation theorems in modal logics. Sufficient conditions . Algebra i Logika , vol. 19 (1980), no. 2, pp. 194–213. English translation in: Algebra and Logic, 19(2), 120–132, 1980.Google Scholar
Maksimova, L. L., Absence of the interpolation property in modal companions of a Dummett logic . Algebra i Logika , vol. 21 (1982), no. 6, pp. 690–694. English translation in: Algebra and Logic, 21(6), 460–462, 1982.Google Scholar
Maksimova, L. L., The Lyndon interpolation theorem in modal logics , Mathematical Logic and the Theory of Algorithms (S. L. Sobolev, editor), volume 2 of Trudy Instituta Matematiki, “Nauka” Sibirsk. Otdel., Novosibirsk, 1982, pp. 45–55.Google Scholar
Maksimova, L. L., Amalgamation and interpolation in normal modal logic . Studia Logica, vol. 50 (1991), nos. 3–4, pp. 457–471.Google Scholar
Maksimova, L. L., The Lyndon property and uniform interpolation over the Grzegorczyk logic (in Russian) , Rossiĭskaya Akademiya Nauk. Sibirskoe Otdelenie. Institut Matematiki im. S. L. Soboleva. Sibirskiĭ Matematicheskiĭ Zhurnal , vol. 55 (2014), no. 1, pp. 147–155. English translation in: Siberian Mathematical Journal, 55(1), 118–124, 2014.Google Scholar
Marx, M., Interpolation in modal logic , Algebraic Methodology and Software Technology (Amazonia, 1999) (A. M. Haeberer, editor), volume 1548 of Lecture Notes in Computer Science, Springer, Berlin, 1999, pp. 154–163.Google Scholar
Pitts, A. M., On an interpretation of second-order quantification in first-order intuitionistic propositional logic . The Journal of Symbolic Logic , vol. 57 (1992), no. 1, pp. 33–52.Google Scholar
Schumm, G. F., Interpolation in and some related systems . Reports on Mathematical Logic , vol. 6 (1976), pp. 107–109.Google Scholar
Schütte, K., Der Interpolationssatz der intuitionistischen Prädikatenlogik . Mathematische Annalen , vol. 148 (1962), pp. 192–200.Google Scholar
Shimura, T., Cut-free systems for some modal logics containing . Reports on Mathematical Logic , vol. 26 (1992), pp. 39–65.Google Scholar
Visser, A., Uniform interpolation and layered bisimulation , Gödel ’96 (Brno, 1996) (P. Hájek, editor), volume 6 of Lecture Notes Logic, Springer, Berlin, 1996, pp. 139–164.Google Scholar